Derivative Calculator Guide: Master Calculus Derivatives Step-by-Step

Derivatives are the foundation of calculus, measuring how things change. Whether you're calculating velocity, optimizing profit, or understanding rates of change, derivatives are essential. This complete guide shows you all the derivative rules with clear examples using MathAI GPT.

1) What is a Derivative?

A derivative measures the instantaneous rate of change of a function. It tells you how fast y changes when x changes—the slope of the curve at any point.

Derivative Notation

All these mean the same thing—derivative of f with respect to x:

f'(x) — Lagrange's notation
df/dx — Leibniz's notation
dy/dx — if y = f(x)
Df(x) — operator notation

2) The Power Rule (Most Important!)

The power rule is the most commonly used derivative rule. For any power of x:

If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹

Bring down the exponent, then reduce the exponent by 1

Examples:

f(x) = x³ → f'(x) = 3x²
f(x) = x⁵ → f'(x) = 5x⁴
f(x) = x → f'(x) = 1 (since x = x¹)
f(x) = 1 → f'(x) = 0 (constant rule)
f(x) = 4x² → f'(x) = 8x (multiply by constant)

3) The Product Rule

When multiplying two functions, use the product rule:

If f(x) = u(x)·v(x), then f'(x) = u'v + uv'

"First times derivative of second, plus second times derivative of first"

Example: Find derivative of f(x) = x²·sin(x)

Let u = x², v = sin(x)

u' = 2x, v' = cos(x)

Solution: f'(x) = 2x·sin(x) + x²·cos(x)

4) The Quotient Rule

When dividing two functions, use the quotient rule:

If f(x) = u(x)/v(x), then f'(x) = (u'v - uv') / v²

"Low d-high minus high d-low, over the square of what's below"

Example: Find derivative of f(x) = x² / (x+1)

Let u = x², v = x+1

u' = 2x, v' = 1

Solution: f'(x) = (2x(x+1) - x²(1)) / (x+1)²

= (2x² + 2x - x²) / (x+1)² = (x² + 2x) / (x+1)²

5) The Chain Rule

For composite functions (function inside another function), use the chain rule:

If f(x) = g(h(x)), then f'(x) = g'(h(x))·h'(x)

"Derivative of outside times derivative of inside"

Example: Find derivative of f(x) = (x² + 1)⁵

Outside function: g(u) = u⁵, g'(u) = 5u⁴

Inside function: h(x) = x² + 1, h'(x) = 2x

Solution: f'(x) = 5(x² + 1)⁴ · 2x = 10x(x² + 1)⁴

6) Common Derivatives to Memorize

Essential Derivative Formulas

Basic Functions:

d/dx[c] = 0
d/dx[x] = 1
d/dx[xⁿ] = nxⁿ⁻¹

Trigonometric:

d/dx[sin x] = cos x
d/dx[cos x] = -sin x
d/dx[tan x] = sec² x

Exponential & Logarithmic:

d/dx[eˣ] = eˣ
d/dx[ln x] = 1/x
d/dx[aˣ] = aˣ ln a

Special:

d/dx[√x] = 1/(2√x)
d/dx[1/x] = -1/x²

7) Real-World Applications

Physics: Velocity & Acceleration

If position is s(t), then:

Velocity: v(t) = s'(t) — first derivative of position
Acceleration: a(t) = v'(t) = s''(t) — second derivative of position

Example: Position is s(t) = -5t² + 20t + 2 meters

Velocity: v(t) = -10t + 20 m/s

Acceleration: a(t) = -10 m/s² (gravity!)

Economics: Marginal Analysis

If cost is C(x), the marginal cost is C'(x)—the cost of producing one more unit.

Similarly for revenue R(x) and profit P(x).

Optimization Problems

To find maximum or minimum values:

Find the derivative f'(x)
Set f'(x) = 0 and solve for critical points
Use second derivative test: f''(x) > 0 = minimum, f''(x) < 0 = maximum

Example: Maximize area of rectangle with perimeter 100

Area: A(x) = x(50-x) = 50x - x²

Derivative: A'(x) = 50 - 2x

Set equal to 0: 50 - 2x = 0 → x = 25

Solution: Square with sides 25 gives maximum area = 625

8) Common Mistakes to Avoid

⚠️ Watch Out For:

Product rule confusion: (uv)' ≠ u'v', use u'v + uv'
Chain rule forgotten: Always multiply by inner derivative!
Quotient rule sign error: It's (low·d-high) - (high·d-low), watch the minus sign
Power rule with coefficients: Don't forget to bring down n AND multiply by existing coefficient
Trig derivatives: d/dx[cos x] = -sin x (note the minus!)

9) Quick Reference: All Major Rules

Derivative Rules Cheat Sheet

Power Rule:

d/dx[xⁿ] = nxⁿ⁻¹

Constant Multiple:

d/dx[cf(x)] = c·f'(x)

Sum Rule:

d/dx[f + g] = f' + g'

Product Rule:

d/dx[uv] = u'v + uv'

Quotient Rule:

d/dx[u/v] = (u'v - uv') / v²

Chain Rule:

d/dx[f(g(x))] = f'(g(x))·g'(x)

10) Practice Problems

Try finding these derivatives, then check with our calculator:

f(x) = 3x⁴ - 2x³ + 5x - 7 (Answer: 12x³ - 6x² + 5)
f(x) = x·eˣ (Answer: eˣ + xeˣ)
f(x) = sin(3x) (Answer: 3cos(3x))
f(x) = (x² + 1)³ (Answer: 6x(x² + 1)²)
f(x) = x/cos(x) (Answer: (cos x + x sin x)/cos² x)

11) Higher-Order Derivatives

You can take derivatives multiple times:

First derivative f'(x) or dy/dx: rate of change
Second derivative f''(x) or d²y/dx²: rate of rate of change (e.g., acceleration)
Third derivative f'''(x): less common (e.g., "jerk" in physics)

Example: Find first and second derivatives of f(x) = x⁴

First: f'(x) = 4x³

Second: f''(x) = 12x²

Get Instant Derivative Solutions

Need to calculate derivatives quickly? Try our Derivative Calculator for:

Any derivative with step-by-step solutions
All rules applied automatically
Partial derivatives for multivariable functions
Higher-order derivatives
Simplification and verification

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